Discussion Paper/Document d‘analyse
2009-13
Network Analysis and Canada’s
Large Value Transfer System
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Bank of Canada Discussion Paper 2009-13
December 2009
Network Analysis and Canada’s
Large Value Transfer System
by
Lana Embree and Tom Roberts
Funds Management and Banking Department Bank of Canada
Ottawa, Ontario, Canada K1A 0G9 [email protected] [email protected]
Bank of Canada discussion papers are completed research studies on a wide variety of technical subjects relevant to central bank policy. The views expressed in this paper are those of the authors.
No responsibility for them should be attributed to the Bank of Canada.
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Acknowledgements
We would like to thank Meyer Aaron, Nikil Chande, James Chapman, Jonathan Chiu, and Miguel Molico for useful comments and suggestions.
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Abstract
Analysis of the characteristics and structure of a network of financial institutions can provide insight into the complex relationships and interdependencies that exist in a payment, clearing, and settlement system (PCSS), and allow an intuitive understanding of the PCSS‘s efficiency, stability, and resiliency. The authors review the literature related to the PCSS network and describe the daily and intraday network structure of payment activity in the Large Value Transfer System (LVTS), which is an integral component of Canada‘s financial system. The picture that emerges confirms that the LVTS is highly centralized among a few key participants, similar to other PCSSs, based on the network of relationships. This could heighten the systemic importance of these participants, and the susceptibility of the system to financial contagion. There have been small variations in the relative importance of individual banks, but the network-wide pattern of
relationships has remained stable from 2004 to 2008, even through the credit crisis.
JEL classification: D85, G10
Bank classification:Payment, clearing, and settlement systems; Financial stability
Résumé
L‘analyse des caractéristiques et de la structure d‘un réseau d‘institutions financières peut jeter un éclairage intéressant sur les relations et interdépendances complexes qui existent au sein d‘un système de paiement, de compensation et de règlement (SPCR), et permettre de comprendre intuitivement l‘efficience, la stabilité et la résilience d‘un tel système. Les auteurs passent en revue la littérature se rapportant aux réseaux de SPCR et décrivent la structure en réseau des opérations de paiement journalières et intrajournalières du Système de transfert de paiements de grande valeur (STPGV), qui fait partie intégrante du système financier canadien. Le tableau qui se dégage de leur étude confirme que, d‘après le réseau des relations, le STPGV est fortement centralisé autour de quelques participants clés, à l‘exemple d‘autres SPCR. Cette configuration pourrait accentuer l‘importance systémique des participants clés et la vulnérabilité du système à la contagion financière. L‘importance relative des banques individuelles a varié légèrement entre 2004 et 2008, mais le profil des relations dans l‘ensemble du réseau est demeuré stable, même pendant la crise du crédit.
Classification JEL : D85, G10
Classification de la Banque : Systèmes de paiement, de compensation et de règlement; Stabilité financière
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Introduction
In financial systems, the network of relationships between financial institutions (FIs) could have important policy implications in terms of counteracting vulnerability to financial shocks and contagion. Central banks and regulators may have disregarded the network aspects of the financial system, even as individual FIs adopted homogeneous business and risk-management practices that resulted in a lack of diversity in the industry, possibly contributing to the fragility that the 2007–09 credit crisis exposed (Haldane 2009). The systemic importance of key participants, which is a critical factor for shocks and contagion, also arguably depends on their interconnectedness in the financial
network. This was the view held for AIG, for which the U.S. Federal Reserve engineered a public buyout in September 2008 (Financial Times 2008).1 Payment, clearing, and settlement systems (PCSSs), in particular, are an integral component of the financial system; they represent the platform on which the last stages of a financial transfer are completed. The PCSS network structure thus offers a relevant and timely perspective on broader financial relationships.
This paper reviews the literature on networks, emphasizing how it might relate to PCSSs, particularly Canada‘s Large Value Transfer System (LVTS); we then empirically
interpret and describe the LVTS as a network of FIs connected by payment flows. Network analysis gives us a way to study and understand the complex arrangement of relationships, and interdependencies, that exist in a PCSS. Examining the network characteristics of a payments system can give us an intuitive understanding of the trade-off between efficiency and stability, which is a prominent concern.
The literature review covers three streams: the first describes other PCSS networks, which we contrast with the LVTS. The second stream explores how network characteristics can influence PCSSs when they face shocks or different system
parameters. For example, theoretical work specifically on PCSSs that relates to network relationships has examined payment coordination and payment queuing. The third stream of literature models network formation. The identification of algorithms that replicate observed structures might offer insight into the plausibility of factors that influence formation.
The empirical section uses an approach similar to that found in the recent literature of Soramäki et al. (2006), Inaoka et al. (2004), and Boss et al. (2003), which aims to describe financial systems through quantitative network analysis. We describe stylized facts of the LVTS payment flow network over the April 2004 to December 2008 period;
1 AIG had sold a large amount of credit-default swaps on fixed-income securities tainted by the
subprime-mortgage-lending crisis, and had an extensive global network ofcounterparties (Walsh and de la Merced 2008).
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the picture that emerges confirms that the structure of relationships in the LVTS is highly centralized among a few key participants. This type of hub-and-spoke structure is similar to other PCSSs. Its small-world structure could increase susceptibility to financial
contagion, while having implications for resiliency, efficiency, and adaptability to shocks.2 The network-wide pattern of relationships has been stable, even if there have been small changes in the relative importance of individual banks. An econometric analysis does not give conclusive results for the influence of the credit crisis on network structure.
Section 2 provides a brief summary of the network concepts and measures used in this study. Section 3 reviews the literature. Section 4 describes our data and their limitations, and section 5 the empirical methodology used. Section 6 describes the results. Section 7 discusses our findings and suggests further empirical research. Section 8 offers some conclusions.
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Summary of Network Concepts and Measures
A network can be described as a structure of nodes connected by links. In the context of a payments system such as the LVTS, nodes represent FIs, and links can represent payment flows, bilateral credit limits, loans, or other variables describing a relationship between two FIs over a given observation period.3 Social network analysis involves different terminology than economists commonly use, so we will begin by defining the main terms and measures used in this study.4 These network measures describe characteristics of PCSSs that might help us to understand, for example, the resiliency and efficiency of the network.
Figure 1 depicts a network between four nodes in both matrix and graphical form. In graphical form, an arrow represents a directed link going from one node to another.5
Undirected links can also describe network relationships; however, we will focus on the
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The LVTS eases the implementation of Canadian monetary policy, and FIs use the LVTS to transfer funds and settle payments related to the delivery of financial assets. These transactions enable the allocation of capital, management of risk exposure, and conversion of savings into investments, which are essential for economic growth. In 2008, about 22,593 payments with a total value of Can$181.6 billion, on average, flowed through the LVTS each business day; see Arjani and McVanel (2006) for an overview of the LVTS. International organizations such as the Bank for International Settlements have recognized the LVTS as being liquidity efficient and comparable in safety to the more traditional real-time gross settlement (RTGS) systems (CPSS 2005, 36).
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The technological infrastructure underlying the LVTS is a static network itself, apart from system outages and the entry of State Street on 18 October 2004. The network of payment flows, however, is more
dynamic, and displays variations according to the time of day and over the calendar year.
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Wasserman and Faust (1994) and Jackson (2006) also provide introductions to the measures used in social network analysis.
5 In PCSSs, directed links are used to represent, for example, which node is the payment sender or credit
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directed case, where links are not necessarily reciprocal. A matrix element mij= 1 represents a directed link. Taking the example of payment flows in a PCSS, mij= 1 represents the occurrence of a payment from sender i to receiver j. The case of mij= 0 represents an absence of any payment flow from i to j for that observation period. The appendix more formally describes how network statistics can be calculated from this matrix.
Figure 1: An Example Network Matrix and Graph
Figure 1 shows a network of unweighted links, where a link between two nodes either exists or does not; a weighted link would indicate the intensity of the link. For example, in a PCSS, a network of weighted links could represent the value of the payments that participants make to each other or a transition probability matrix for liquidity movement between banks.6 Many network measures simplify relationships to unweighted links, as in Figure 1, which is a reasonable place to start.
Some of the basic network measures, such as connectivity, reciprocity, and degree, relate to a counting of links. Connectivity calculates the number of actual links that exist in the network divided by the total number of potential links, giving a measure of network completeness. In a PCSS, connectivity is a measure of activity and indicates whether participants are transacting, in general, with many or only a few of the other participants. In Figure 1, there are 8 observed links and 12 potential links; thus, connectivity is 2/3. For directed graphs, reciprocity measures the percentage of observed links that also have a link travelling in the opposite direction; as the word implies, this is the percentage of links that are reciprocated. In Figure 1, A and D have a reciprocal relationship and C and D have a reciprocal relationship; thus, four links are reciprocated. There are eight
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In Bech, Chapman, and Garratt (2008), a transition probability matrix reflects the probability that a dollar will be sent from each LVTS participant to any other participant. They use the stationary equilibrium resulting from this matrix to identify the central participant in terms of who is likely to hold the most liquidity. A C 0 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 A B C D A B C D B D O U T IN
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observed links in Figure 1, so the reciprocity in this example is 1/2. In the LVTS, reciprocity could provide information about payment coordination, which influences liquidity needs. For example, if there is a reciprocal relationship between two
participants, one of the participants is using a payment received from the other as a source of liquidity to send a return payment, rather than drawing on its bilateral credit limit (BCL).7
Degree is a count of a node‘s direct relationships, also known as its neighbours. The
in-degree of node i is the number of links that end at i, while the out-degree of i is the number of links that start with i. For example, in Figure 1, node D has an in-degree of 2 and an out-degree of 3. In a PCSS, degree indicates the number of counterparties that a participant interacts with during a given period.
The degree distribution is afrequency distribution that describes how many nodes have
any given degree in the population of nodes. Figure 2 depicts the out-degree distribution of the example network in Figure 1. Empirical degree distributions often seem to
resemble a power-law distribution, both in the natural world and the social sciences. For example, Barabási and Albert (1999) find power-law distributions for article citations in the scientific literature, and for the numbers of links to web pages. Power-law
distributions are scale free – for a given ratio of two values in the distribution, the relative frequency of encountering the two values does not change. For example, with a power-law exponent of 2, a node of degree 6 is four times less frequent than a node of degree 3; a node of degree 10 is four times less frequent than a node of degree 5. Power-law networks have a few degree nodes, and many nodes of low degree, although high-degree nodes are more common than in a Gaussian distribution. Soramäki et al. (2006) show that, in some fields, scale-free networks are robust to the removal of random nodes but are vulnerable to becoming disconnected by the removal of highly connected nodes. However, they also note that the structure of a network and the mechanisms for contagion differ across networks and fields and therefore the vulnerability to targeted node removal might not necessarily apply to PCSSs. They note that, in a PCSS, understanding how liquidity flows through the system will help to assess the robustness of the network.
7 In the context of a PCSS, liquidity is the ability to meet payment obligations, by having funds or credit
5 Figure 2: Out-Degree Distribution of Figure 1
The clustering coefficient measures the proportion of a node‘s neighbours that are also
neighbours to one another. In other words, it measures the probability that if node i is linked to nodes j and k, then j and k are also linked. In Figure 1, node C has two
neighbours, B and D.8 Two links can exist between B and D: a link from B to D and one travelling in the opposite direction. One of these possible links exists in the example graph; therefore, node C‘s clustering coefficient is 1/2. A clustering coefficient for the network averages these coefficients across individual nodes. The network clustering coefficient in Figure 1 is 1/2.9
Paths help to measure how close nodes are to one another at any given time. A path is a
sequence of nodes and links beginning and ending with nodes, where any link or node is not included more than once. The length of a path is measured by its number of links. An example path in Figure 1 begins at node A, goes through node D and then ends at node B, for a length of two. In the LVTS, all participants can make payments to all others, so paths do not reflect the course that payments can travel; however, a path can reflect the course that liquidity or contagion could follow.10
8 The clustering coefficient is often defined for out-links of nodes. Using that approach, nodes B and D are
C‘s neighbours because a link travels from C to these nodes. Node A is not C‘s neighbour because the link travels from A to C; however, node C is A‘s neighbour. This occurs because Figure 1 is a directed graph, where relationships are not necessarily reciprocal.
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Node A‘s clustering coefficient is 1, while nodes C and D both have clustering coefficients of 0.5 and node B has a clustering coefficient of 0 because it has only one direct neighbour. Soramäki et al. (2006) note that, by definition, a node with degree 1 has a clustering coefficient of zero.
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The distance between a pair of nodes is the length of the shortest path connecting them.
For example, in Figure 1, the distance from node C to A is two. In PCSSs, distance might provide an indication of how a shock affecting one node could travel through links to affect another. The average path length is the average distance for any combination of two nodes in the network. In Figure 1, the average path length is 1.33.11 In the LVTS, an average path length of one indicates that all participants have sent a payment to all others. A longer path length indicates that activity is concentrated among fewer pairs of
participants. When there is no path between a pair of nodes, that distance is undefined, so we calculate distance-based measures for the giant strongly connected component
(GSCC). The GSCC is the largest subnetwork where all members have both ingoing and outgoing links to the rest of the GSCC as a whole. There is a directed path between any two nodes that are members of the GSCC.
Eccentricity and diameter provide further descriptions of the distance between nodes. The
eccentricity of a node is the distance to the node that is the farthest away. In Figure 1,
node C‘s eccentricity is two, since it is two links away from node A and all other nodes are closer. The diameter is the largest eccentricity in the network, and indicates the maximum distance between any two nodes in the network. In the example in Figure 1, the diameter is two because the distance between all pairs of nodes is two or less. Eccentricity and diameter can provide an indication of how easily or quickly an event affecting one node could potentially affect the other nodes in the network. For example, if one participant ceases to send payments, those with direct relationships might find
themselves short of liquidity sooner than those who have only indirect relationships with that participant.
Centrality helps to indicate the importance of a given node to the network as a whole. We
measure degree, value, and betweenness centrality.12 For PCSS studies, centrality would ideally identify systemically important participants, which for example might be
important for the smooth flow of liquidity through the system.13 Also, a disruption or default might have a larger impact on the network if a central participant experienced the disruption or default.
Degree centrality equates centrality directly with the degree of a node and hence does not
consider indirect relationships. Applying this measure, the most central node is the one with the most direct relationships. For this paper, we have defined value centrality to mean a node‘s share of the total gross value of payments sent and received, which also
11 In the example in Figure 1, there are eight pairs of nodes that have a path length of 1 and four pairs of
nodes with a path length of 2. Thus the average path length is 16/12 or 1.33.
12Other centrality measures are discussed in Wasserman and Faust (1994), Jackson (2006), and Borgatti
(2005).
13 von Peter (2007) uses centrality to identify international banking centres.
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considers only direct relationships. A bank that was a counterparty to all transactions would have the maximum centrality for either degree or value centrality.
Betweenness centrality considers both direct and indirect relationships. The betweenness
centrality of a node is the probability that the node is used as an intermediary on the shortest path between any two other nodes; it aims to measure how important a node is for flows between other nodes in the network. Different centrality measures could allow different patterns to become apparent, even though they are based on the same links. In the case of LVTS payment flows, centrality measures have strong positive correlations.
Centralization provides a summary statistic of the extent to which the network is
centralized; it takes the centrality scores of individual nodes and sums up their deviations from the maximum centrality score. Centralization is normalized by the ―theoretical maximum deviation‖ to give a value in the range of zero to one: zero represents a network where all nodes have equal centrality, and one represents a hub-and-spoke network where a single node has maximum centrality and all other nodes are equally peripheral. Centralization could be useful for comparing across networks or across time. Figure 3 shows a decentralized network on the left, where all nodes have links to all other nodes. Using any of the centrality measures, the nodes are all equally central, and the centralization value of this network is zero. In the hub-and-spoke network on the right, the hub node is central, and this network has a centralization value of one.
Figure 3: Two Example Networks
A caveat to these statistics is that they do not incorporate the varying importance of individual links, for example, by reflecting payment values or transition probabilities. The distribution of individual transaction payment values is highly skewed to the right, with the majority of payments having relatively low value; a later section will discuss the similar case of bilateral links. Meanwhile, the bulk of value that FIs transfer through the
A C B D A C B D Decentralized network Centralized hub-and-spoke network
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LVTS occurs in the smaller number of high-value payments.14 A possible extension of this analysis is to formulate new measures that suitably capture the importance of links relative to the size of an FI. Value centralityis one measure that simply reports on a bank‘s share of gross payment flows.
Neither do network statistics of payment activity represent financial exposures, which would be important for financial contagion. But a network analysis of payment flows could be valuable as a tool to understand broad patterns of activity, when detailed data of financial exposures are not available or manageable: it would reveal which participants play an important role, and the avenues for liquidity flows, which relate to financial contagion.
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Related Literature
Network analysis first appeared in statistical physics and sociology; the existence of network-like structures is common in both the natural and human world. Examples include river systems, the Internet, and networks of human relations (Inaoka et al. 2004). Though the network literature is multidisciplinary, we focus on economics papers and other papers that could be relevant for understanding PCSSs.15
We cover three streams of literature: the first includes a small number of empirical papers that have used social network measures to study financial system networks. Empirical results give a reference point for theoretical work, and allow us to compare the network structure of the LVTS to other PCSSs. The second stream of literature comprises theoretical models of PCSSs and examines how network characteristics of the system influence outcomes. The third stream models network formation; these models are not necessarily found in the economics literature, but we discuss their potential applications to PCSSs. Models of network formation can contribute to our understanding of why certain network structures arise under different conditions, which could permit insight into participant behaviour.
3.1
Empirical network studies of PCSSs
Soramäki et al. (2006) describe the topology of interbank payment flows between participants in the Fedwire Funds Service, which is the large-value payment system (LVPS) at the core of the U.S. financial system; Inaoka et al. (2004) describe payment flows in BOJ-NET, which is Japan‘s LVPS; Boss et al. (2003) examine the network of
14 In 2008, the average LVTS payment value was $8.04 million, while the median was $44.6 thousand. The
extreme values for a single payment were close to $13.4 billion, and less than a dollar. Fedwire had an average of $3 million and a median of $30 thousand for the first quarter of 2004.
15 Jackson (2006) summarizes and discusses the economic applications of network analysis. Haldane (2009)
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interbank loans in Austria.16 Figure 4 depicts these networks, and Table 1 summarizes some of their results, along with the corresponding measures for the LVTS. In contrast to the usual network algorithms in the literature, Bech, Chapman, and Garratt (2008) use Markov theory to identify central participants in the LVTS, and Chapman and Zhang (2009) identify communities within the LVTS, based on the probability of different bank interactions.
In Figure 4, points or circles represent banks, and lines represent payment activity between those banks. Fedwire has over 9.5 thousand participants, of which about 6.5 thousand might be active on a given day; therefore, in the top left, we can see only the basic shape of a dense core of the 1 January 2004 network, along with the rest of the network, which becomes increasingly less connected, away from that core. The core of Fedwire in the top right is the part of the payments system that accounts for 75 per cent of the value transferred. We can see about four banks that appear to be more strongly
connected than others. Network measures might have further identified their relative systemic importance, possibly contributing to different policy prescriptions for those banks during a crisis.
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Table 1: Comparison of International Network Structures LVTS payments Fedwire payments Austrian interbank loans BOJ-NET payments Time period Daily.
April 2004– December 2008.a Daily. First quarter 2004. Quarterly single-month periods between 2000 and 2003. Monthly. June 2001.
Payment statistics, U.S.-dollar total for 2004b
Value (trillion)c 25.4 478.9 188.8 Volume (million)c 4.4 125.1 5.2 Avg. value (million)c 5.8 3.8 36.5 Value as % of GDPc 2,558 4,099 4,099
Network statistics (variable time frames)
Number of nodesd 14 5,086 ±123 883 354 Number of links 76,614 ±6,151 1,727 Connectivity (%) 69.2 ± 3.3 0.3 ±0.01 2.76 Reciprocity (%) 89.3 ± 2.5 21.5 ±0.03 Average path length 1.31 ±0.03 2.62 ±0.02 2.59 ±0.02 Average eccentricity 1.84 ±0.07 4.67 ±0.33 Average diameter 2.01 ±0.07 6.6 ±0.5 Clustering coefficient (%) 84.3 ± 1.5 53 ±1 12 ±1 Power-law exponent 2.11 ±0.01 2.15 ±0.01e 2.01f 2.1
a. The LVTS measures use a threshold of 1 per cent of the lower Tranche 2 Net Debit Cap (T2NDC) for each bilateral pair of banks, averaged over the study period. A participant‘s T2NDC is the limit on their multilateral net debit position, and relates to the bilateral credit limits granted to it by other participants. See section 5 for further details.
b. The average Can$/US$ exchange rate for 2004 was 1.3012 (CPSS 2008). c. CPSS 2008.
d. The number of nodes in BOJ-NET is the number of banks for which there are 21 or more
transactions with at least one other bank over the 1-month period studied. The number of nodes in Fedwire on a given day is the number of nodes that are part of that day‘s GSCC. The number of nodes in Boss et al. (2003) is the number of banks. The number of nodes in the LVTS excludes the Bank of Canada.
e. The first power-law exponent listed in the Fedwire study corresponds to the out-degree distribution, and the second to the in-degree distribution.
f. The power-law exponent reported for this study is for undirected links. When considering directed links, the results are 1.73 for in-degree and 3.11 for out-degree.
Sources: Inaoka et al. (2004), Soramäki et al. (2006), Boss et al. (2003).
Network analyses of payments systems typically find that a small number of FIs are highly connected in their payment flows, and thus act as hubs for the system. Most FIs have much fewer links, and only a small proportion of potential links in the network
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actually exist. For Fedwire, the connectivity of the GSCC is 0.3 ±0.01 per cent, while in BOJ-NET connectivity is 2.76 per cent. Despite low connectivity, the path length
between any two FIs is often short due to the presence of hubs. The average path length for Fedwire is 2.62 ±0.02, while for loans in the Austrian interbank system it is 2.59 ± 0.02.
Figure 4: International LVPS Networks
Fedwire interbank payment network. Fedwire core accounting for 75 per cent of value 1 January 2004 (Soramäki et al. 2006). transferred. 1 January 2004 (Soramäki et al. 2006).
BOJ-NET (Inaoka et al. 2004). The banking network of Austria (Boss et al. 2003). Figure 5 shows the 14 LVTS participants apart from the Bank of Canada, with average daily payment flows ranging from thin pink lines, for small values of Can$10 million or less, to thick black lines for larger values. The figure illustrates some general points about the LVTS network of average payment activity in 2008: (i) even a small payments system like the LVTS would be difficult to describe and understand without the right approach; (ii) the LVTS has high connectivity of around 69.2 ±3.3 per cent on a daily basis; (iii) there is considerable heterogeneity in the strength of links, and a participant‘s
interconnectedness in the LVTS might not correspond exactly to its size; (iv) much like what we can gather from the picture of Fedwire‘s core in Figure 4, this figure suggests
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that the systemic importance of a participant depends partly on its interconnectedness, with only a small number of banks that are heavily connected with thick lines.
Figure 5: LVTS Average Daily Gross Payment Flows, 2008
The average path length in the LVTS is 1.31 ±0.03; despite Fedwire‘s much lower connectivity, its average path length is only twice that of the LVTS, because of the role of network hubs. In the end, we could describe all of these systems as small worlds, which exhibit high clustering and short distances between banks, even when the prevalence of direct relationships is low. These short distances can mean that it takes only a small number of steps for liquidity to circulate between any two banks, or for disruptions to spread.
In Fedwire, half of the nodes have four or fewer outgoing links, while the maximum degree for a node reaches 1,922 ±121 (out-degree) and 2,097 ±115 (in-degree), with over 5,086 ±123 participants active on a given day. The degree distribution for Fedwire follows a power-law distribution, for degrees greater than 10, which is similar for both in- and out-degrees. Maximum-likelihood estimates of the power-law exponents are 2.15 ± 0.01 for in-degree and 2.11 ±0.01 for out-degree (Soramäki et al. 2006). Boss et al. (2003) find a power-law exponent of 2.01 for undirected links, while Inaoka et al. (2004) estimate an exponent of 2.1 and find a fractal structure similar to other network structures such as river basins or the Internet. The smaller size of the LVTS does not afford much opportunity for such an assessment, but, for a threshold where we examine only links that have a value of at least 40 per cent of the lower Tranche 2 Net Debit Cap (T2NDC) for a given pair of banks,17 the big five banks have an average degree over three times greater than the other banks. These hubs would seem to have the greatest potential for
propagating financial contagion, because of the network‘s reliance on their interconnecting role.
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Networks with power-law distributions in other disciplines have been found to be robust to random failures, but not to a failure of a hub (Albert, Jeong, and Barabási 2000; Crucitti et al. 2004). Whether this result also holds for payments systems depends on the mechanisms of contagion and adaptation by FIs (Soramäki et al. 2006). Nevertheless, the network structure of payments systems suggests that they have evolved primarily with efficiency in mind, rather than resiliency to extreme events (Inaoka et al. 2004). Some banks would presumably take on the role of hubs because of efficiency advantages, whether this was to result from economies of scale, information asymmetries, history, or other reasons. An ability to intermediate information and liquidity flows could be a further advantage for a hub bank, such that a network structure is self-reinforcing, much like in the preferential attachment models we will discuss later. Conversely, inefficiencies or disadvantages that impede the growth of smaller institutions in the network might have a similar self-reinforcing effect.
A complete network might create resilience through risk dispersion by spreading the impact of a shock safely across many nodes (Allen and Gale 2000), but, at the same time, it could increase vulnerability to shocks that exceed a threshold beyond which the system is no longer self-repairing (Haldane 2009). Increased financial network complexity, partly due to structured financial products, can amplify uncertainty because the precise source and location of underlying claims becomes unknown. If there is also homogeneity in business and risk-management practices, a relatively small initial financial disturbance could result in herd behaviour and a cascade of losses that transmits easily through a highly connected network (Haldane 2009).
Fedwire reciprocity is low on any given day, at around 22 ±0.3 per cent, and is
uncorrelated with the number of nodes and links, or the volume and value of payments. The clustering coefficient is 53 ±1 per cent, which is high relative to the clustering of Austrian interbank loans at 12 ±1 per cent. Clustering in the Austrian system might be relatively low because the cost of link formation for smaller Austrian branch banks, which mostly rely on a head institution, might not justify the benefits of diversification (Boss et al. 2003). Daily reciprocity and clustering in the LVTS are high, at around 89.3 ±2.5 and 84.3 ±1.5. The tendency for reciprocity to be high relative to connectivity reflects the bilateral coordination of payments and its importance for managing liquidity. More involved algorithms have further identified network characteristics of the LVTS. Bech, Chapman, and Garratt (2008) apply Markov theory to identify the central
participants in the LVTS. In that paper, the central participant is the one that we expect to hold the most liquidity, where the network structure represents a transition probability matrix of liquidity movement that determines liquidity holdings. Chapman and Zhang (2009) use transactions-level data to identify where LVTS participants transact more heavily with other participants in the same community than with those in different
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communities. Using two different measures of transaction intensity, the authors find that there could be communities within the LVTS based on either transaction size or the participants‘ geographical locations.18
Papers that characterize an existing network are an important start to further research on PCSSs. Network descriptions can provide input for modelling how networks form, which we discuss in section 3.3. In addition, there is evidence that network structure can
influence outcomes, as shown in the models we describe in the next section. Descriptive studies of PCSSs can contribute to the discussion and intuitive understanding of observed outcomes.
3.2
Models of PCSSs
Existing papers that incorporate a network into PCSS modelling take the network as given. These models help researchers understand how the existing network influences economic outcomes; the models do not attempt to explain why a certain network exists. In this section, we describe several papers: Bech and Garratt (2006) examine disruptions, Allen and Gale (2000) study a liquidity shock and whether it leads to contagion, Gale and Kariv (2007) examine the network of trading relationships and what features can lead to market breakdowns, and Martin and McAndrews (2007) model payment queuing.19 These papers demonstrate that network characteristics can influence economic outcomes, and that such networks can be complementary to economic modelling.
Bech and Garratt (2006) study the equilibrium effects of a disruption in the ability of a few participants to send payments, and explore the impact of the network on
coordination. They use a game-theoretic model with a large bank and several small banks. Banks choose to send payments in the morning or the afternoon, depending on the cost of delay and liquidity. When each bank chooses whether to make payment in the morning or afternoon, they consider the action taken by the large bank, and the number of small banks making their payments in the morning, since this influences the cost of liquidity. Banks obtain liquidity in part from incoming payments; if there is a substantial shift in payments to the afternoon, a bank might also prefer to pay in the afternoon rather than face the costlier alternative of intraday credit to fund a morning payment.
The model begins in the efficient equilibrium where all participants pay in the morning; a disruption then temporarily forces a subset of banks to pay in the afternoon. An
alternative equilibrium occurs if all participants pay in the afternoon; this equilibrium is
18 Chapman and Zhang (2009) note that geography itself might not be the cause of the community, but
rather that geography might serve as a proxy for other characteristics or business relationships that lead to the formation of the community.
19 The Bech and Garratt (2006) and Martin and McAndrews (2007) models are both based on the
15
inefficient because of delay costs. The main question that Bech and Garratt (2006)
address is whether the disruption will cause the banks to move to the inefficient afternoon equilibrium. The outcome depends on the decisions taken by each bank, and in particular the large bank, since the incentives each bank faces to move to one equilibrium or the other depend on whether the other participants are paying in the morning or the afternoon.
The network of the system plays an important role: when the large bank is bigger, or when small banks do not often transact among themselves, the large bank influences the outcome more. If the small banks transact only with the large bank, we observe a hub-and-spoke network, with the large bank in the centre; if small banks also transact among themselves, we observe a more complete network. It is not surprising that the influence of the large bank is greater in the hub-and-spoke network, because small banks are entirely reliant on the large bank for any incoming liquidity that they receive.
Allen and Gale (2000) examine the role of the network between banks in financial contagion. In this model, banks use an interbank market to redistribute liquidity among themselves. When a bank experiences a liquidity shock, the network topology will influence how the shock might spread to other banks. If the network is complete, all banks will experience some loss. Dispersing the loss across all banks can diminish the impact of the crisis on each bank and reduce the likelihood of contagion. Conversely, when the interbank market is incomplete, the immediate counterparty exposure could be concentrated on fewer banks, leading to a greater risk of contagion where additional banks succumb to the shock.
Gale and Kariv (2007) model a market for trade, where an assumed network determines which nodes can trade. In a complete network, all nodes can trade with any other node in the network, whereas if the network is incomplete some nodes cannot trade directly with each other. Using this approach, the LVTS would be a complete network, since all participants can trade with each other, although LVTS participants would be
incompletely connected to domestic or international systems beyond the LVTS. The authors suggest that an incomplete network might occur if asymmetric information leads participants to transact only with participants that they ―know and trust.‖ An incomplete network could also occur where an intermediary acts as central counterparty and
guarantor. Gale and Kariv (2007) show that transaction costs could also lead participants to limit the number of counterparties, or lead small traders to rely on an intermediary broker. In other words, a move to efficiency could result in some nodes taking the role of intermediary in an incomplete network. The authors argue that market breakdowns can occur if the network structure changes, representing changes to which participants can trade and act as intermediaries.
16
While the issue of financial contagion comes to the forefront during most financial crises, the efficiency and reliability of operations is another concern for payments systems. The arrangements that determine when queued payments settle provide one example. Martin and McAndrews (2007) model two different queuing mechanisms and study how the network of payments influences which queue generates the highest participant welfare. In their model, participants decide whether to make payments in the morning, the afternoon, or to place them in a queue in the morning.
The network of payments, and how they offset, determines whether payments are released from the queue. Martin and McAndrews (2007) examine two network types. In the first network, all payments offset bilaterally. In this case, a payment in the queue is released if the offsetting payment is also in the queue or if the offsetting payment is made in the morning. In the second network, all payments form a circle. In this case, a payment in the queue can be released only if the sender has received a payment in the morning, or is the recipient of a queued payment that is released by a payment made in the morning. The network type influences the probability of settlement for a payment in the queue, and therefore influences the incentive to place payments in the queue and the resulting
welfare that the queue provides. Other parameters, such as the probability of time-sensitive payments and of liquidity shocks, also influence queue usage and welfare. Many models do not incorporate agent behaviour, which would adapt to changes in policy or other events. Because of the way it aggregates interactions, network analysis can allow for easier interpretation of whether economic shocks result in behavioural changes. One possible approach for payments systems that does consider changes to participant behaviour, and the influence of network effects, is agent-based modelling.20 This approach will be of particular interest for the study of participant responses to policy changes or actions, or disruptions such as operational outages.
The models discussed in this section demonstrate that a network structure can influence economic outcomes in a PCSS. Because of the network‘s effect on the results of shocks and the impact of system parameters, such as the queuing mechanism, it is important to understand the network structure. But these models do not try to explain why a certain network exists. The next section describes models of network formation that do address this question.
3.3
Models of network formation
Theoretical models, in various fields, have offered explanations for how networks form. We describe several of those models in this section, focusing on how they can be used to
20 Work has begun at the Bank of Canada to develop an agent-based model of the LVTS (Labelle St-Pierre
17
better understand PCSS networks. We discuss random formation, preferential attachment, and strategic formation models. Each of these classes of theoretical model can provide insight into how network characteristics might be determined. Additionally, some of the models could provide insight into how the network might change in response to a shock, where these changes reflect participant behaviour.
Models of network formation can help us to understand what network will form under certain conditions and shocks to the system. Models of random network formation assume that links randomly form between a fixed number of nodes. The Erdös-Rényi (1960) model is a basic prototype where each pair of nodes has an unconditional
probability p of being linked. When the number of nodes is large, the degree distribution is approximated by the Poisson distribution (Jackson 2006).
Random models have been extended, or adapted, to account for the observed clustering and degree distribution found in real-life networks, which the purely random model does not generate. These extensions typically involve non-random elements; for example, Watts and Strogatz (1998) and Newman and Watts (1999) begin with clustering and randomly change or add links, which allow these models to account for observed clustering and short path lengths. Newman, Strogatz, and Watts (2001) model random graphs while imposing an arbitrary probability distribution for the node‘s degrees. An example of how random models could relate to a PCSS is that the times at which LVTS participants receive payment requests from clients could be thought of as the outcome of a random process that would generate a network of payment flows. Because payments are not necessarily submitted to the LVTS when the request is received, the observed LVTS payment times will not perfectly reflect the random process. But Newman, Strogatz, and Watts (2001) suggest that random network models can act as benchmarks for observed networks. The benchmark can help identify characteristics in the observed network that are caused by participant incentives rather than randomness, and researchers can then begin to identify what those incentives might be.
Preferential attachment models also provide an explanation for how networks form. Barabási and Albert (1999) describe the basic design of preferential attachment models. They are typically characterized by growth: each period, a new node joins the network and forms links with existing nodes. In preferential attachment models, nodes are more likely to form links with high-degree nodes than with low-degree nodes. In a network of friendships, for example, a new person would often prefer to befriend a well-connected person, because having many friends-of-friends confers benefits in the form of social encounters, or more potential friends, in addition to whatever intrinsic value the well-connected person might provide (Brueckner 2004). In a financial network, indirect links
18
could increase market access and trading opportunities through counterparty intermediation.
Various model designs can produce preferential attachment; in some models, the
probability of linking with node i is proportional to i‘s degree, as in Barabási and Albert
(1999). Klemm and Eguíluz (2002a, b) create a preferential attachment model where nodes are deactivated based on their degree. When nodes are deactivated, they no longer gain links as new nodes join the network; deactivation could be difficult to interpret for PCSSs. The preferential attachment models result in high-degree nodes acquiring more links than low-degree nodes, which allows these models to produce power-law degree distributions.
Extensions or alterations to the basic preferential attachment models more closely
replicate observed network structures. For example, Klemm and Eguíluz (2002b) model a preferential attachment model where there is a probability that each new link will be randomly changed to link to an inactive node. In a model by Jackson and Rogers (2007), nodes form links randomly but also form additional links with some immediate
neighbours of the randomly encountered nodes. These extensions allow preferential attachment models to produce commonly observed real-world network characteristics, such as small distances, small diameters, and large clustering coefficients.
Preferential attachment models rely on growth in the number of nodes; however, PCSS networks do not generally grow. Preferential attachment models might be useful for PCSSs if they accurately explain the existing network, or if the existing network has features that behave like growth. For example, we might observe networks that resemble growth models if participants move sequentially. Also, preferential attachment networks tend towards a hub-and-spoke type of structure, where many low-degree nodes connect to a high-degree node, similar to what is found in the empirical studies described in section 3.1.
Strategic models also explain network formation. In these models, a payoff function describes the costs and benefits accruing to a node from a given network; thus, links will form only if they provide a strategic benefit. Bala and Goyal (2000) outline a general payoff function that can be adapted to reflect the incentives faced by nodes, such as indirect links creating value (Bala and Goyal 2000; Jackson and Wolinsky 1996), or different costs, depending on which nodes link (Jackson and Rogers 2005). In PCSSs, a participant might face different costs for an interbank loan based on whom they are borrowing from, because the lender has incomplete information about the participant‘s creditworthiness.21
19
When payoffs are modelled explicitly, as in these strategic formation models, we can study the welfare that a network creates, and thus can compare networks based on total welfare; the network that provides the highest total welfare is the socially optimal network. Links create externalities if a node‘s payoff depends on the entire structure of the network. Jackson (2006) explains that externalities mean that the socially optimal network will not be stable if it does not align with individual incentives.
Strategic formation models need to define a transition process that dictates how nodes are permitted to change their links. Bala and Goyal (2000) describe a transition process where nodes can observe the network in the previous period and determine whether to add or sever links. This basic process can be altered; for example, Jackson and Watts (2002) allow just one link to change per period, while Bala and Goyal (2000) allow multiple links to change. Finally, some studies, such as Herings, Mauleon, and
Vannetelbosch (2006) and Page, Wooders, and Kamat (2005), allow for forward-looking nodes. In models with forward-looking nodes, the transition process permits nodes to consider the effect of their choices on the subsequent choices of other nodes.
Strategic formation models also need to specify a stability definition. The stability definition determines when the network will stop changing. A stability definition might be a Nash equilibrium where all nodes play their best response given the actions taken by other nodes. Nash equilibrium is used in Bala and Goyal (2000), whereas pairwise
stability and core stability are modelled in Jackson and Watts (2002). Bloch and Jackson (2006) discuss some of the stability definitions used in models of network formation. Strategic models might be the most useful of the formation models for explaining PCSS networks. Many of the PCSS participants‘ actions are the result of strategic decisions. For example, FIs make strategic decisions when granting BCLs. Participants likely choose a BCL to maximize the benefit of receiving payments, and resulting higher liquidity, subject to the cost of credit exposure and collateral. Similarly, interbank loans are strategic decisions balancing interest earned against credit risk. In addition, the decision regarding when to submit payments could be modelled as a strategic decision weighing the costs and benefits of delaying payments. However, some incentives faced by PCSS participants could be difficult to accurately identify and model.
Some work has begun to model network characteristics as strategic decisions; for example, Chapman, Chiu, and Molico (2008) model the decision to become a direct participant in the payments system or to tier through a clearing agent (CA). In this model, the cost structure and information structure are important determinants of the network. Certain small participants do not have public credit histories. If these participants choose direct participation, their counterparties will not lend to them, since the counterparties cannot assess whether the small participant is a safe debtor. This is inefficient, because of
20
the resulting liquidity costs. These small participants can instead choose to tier under a CA who has a public credit history. If they tier under a CA, the frequent interactions between the CA and the small participant mean that the CA can monitor the small participant‘s creditworthiness and optimally choose whether that participant can borrow. Small participants who are safe debtors will be able to borrow and save liquidity costs when they tier under a CA.
The strategic models are also useful for PCSSs because these models might allow us to better understand how the network will differ if the modelled incentives faced by the participants change, due to either a shock or a change in the system parameters.
4
Data
LVTS data enable us to identify the sender, recipient, payment submission time, amount, and tranche, from April 2004 to December 2008, for 1,196 daily observations.22 The Canadian Payments Association (CPA), which operates the LVTS, provides the data to the Bank of Canada. The fifteen LVTS participants include the six major Canadian banks, four smaller Canadian FIs, four subsidiaries of foreign-based banks, and the Bank of Canada.
There are limitations to the data. The Canadian financial system extends beyond the LVTS such that direct participants act as correspondent banks for other FIs or
organizations. Hence, the data describe only activity passing through a part of the core of the Canadian financial system.23 There is no way to identify the original sender or final intended recipient, in the case of intermediated banking, and the motivation for payment flows is usually unknown.24
In 2008, daily LVTS transaction values averaged around Can$181.6 billion, with daily transaction volumes of around 22.6 thousand. The maximum daily value of
Can$268.6 billion was on 4 September 2007, and the maximum volume of 39,303 was on 2 July 2008. The average transaction value was Can$8.04 million in 2008. Tranche 2 in 2008 accounted for 85.2 per cent of the LVTS value and 98.6 per cent of the volume, and had an average transaction value of Can$6.94 million. During the day, value peaks
22
Participants in the LVTS can send a payment through one of two payment streams. In Tranche 1, they collateralize any credit usage on a dollar-for-dollar defaulter-pays basis, while Tranche 2 uses a survivors-pay collateral system. Tranche 1 accounts for about 10 to 15 per cent of survivors-payment value, and 1.5 per cent of payment volume. LVTS participants typically use Tranche 1 for time-sensitive payments, and payments to or from the Bank of Canada. Tranche 2 economizes on collateral requirements and is thus the main LVTS payment stream (Arjani and McVanel 2006).
23Neither would these data encompass activity netted out in the CDSX and Automated Clearing Settlement
System (ACSS) payments systems.
24 It is possible, though, to filter out certain types of transactions, by matching their expected characteristics
to payments. Furfine (1999) and Hendry and Kamhi (2007) are thus able to characterize the markets for federal funds in Fedwire, or uncollateralized overnight loans in the LVTS, respectively.
21
anywhere between 16:00 and around 17:00, and tapers off by 18:00, with modest bumps closer to midday, as shown in Figure 6.
The number of transactions shows the strongest peaks at 06:00, 07:00, and 08:00 (Figure 6); the first peak, at 06:00, could result from the release of overnight payments at the start of the daily LVTS processing cycle for general payments exchange. On U.S. holidays in the study period, total daily value and volume in the LVTS decline by an average of 61 per cent and 34 per cent, respectively. Civic holidays correspond to an average decline in daily value of 95 per cent from the mean and a decline in transactions volume of 82 per cent. The August civic holiday is observed in all Canadian provinces except Quebec, but banks continue operations to deal with CLS Bank and Quebec-based institutions. KPSS tests suggest that levels or logs of value and volume are trend stationary; transactions volumes have a certain amount of monthly periodicity, from the pattern of
autocorrelations in the linearly detrended component.
The big five banks account for about 75 per cent of the transacted value for both tranches in 2008. A Herfindahl-Hirschman index measure of gross payment flow shares among the participants gives daily values of between 1,495 and 1,876 with a mean of 1,695, which indicates that some banks play a much more prominent role than others do.25 The results of other measures of centralization specific to network analysis are reported later; in contrast, payment shares and the Herfindahl-Hirschman index do not consider network structure.
25
The Herfindahl-Hirschman index is calculated by hh share12 share22 sharen2with shares in percentages, such that 10,000 indicates a pure monopoly.
22
Figure 6: Average Total Value and Transactions in the LVTS, 10 Min. Intervals
5
Methodology
The LVTS is a relatively small payments system, such that a network analysis of daily payment activity has a different character than for other systems; the LVTS network is almost completely connected, and, based on centrality measures, it would be difficult to distinguish between participants, without a threshold. We therefore examine both a daily frequency and intraday sampling frequencies of 5, 10, 15, 20, 30, and 60 minutes, and use thresholds for the daily case.26
The analysis focuses on Tranche 2 transactions, which represent the majority of value and volume in the LVTS, and excludes the Bank of Canada. In Tranche 1, a participant must fully collateralize its debit limit, resulting in different risk characteristics than in Tranche 2, which instead uses a survivors-pay mechanism that would fully collateralize the largest single default. The Bank of Canada relies mainly on Tranche 1 and serves a much
different role relative to the private banks.27
26 In this paper, we use the statistical computing software R and the social network analysis tools (Butts
2009) that are available for that software.
27
The inclusion of Tranche 1 and the Bank of Canada results in connectivity, reciprocity, and clustering that are somewhat lower, reflecting the less-frequent involvement of the Bank of Canada in the payments system. Distance-based measures are not greatly affected, while degree and closeness centralization are higher, which is not surprising if the network has become more sparse.
23
The existence of a link depends on the length of the data sampling frequency and any minimum threshold of payment value that qualifies as a link. Obviously, there is some discretion according to the choice of sampling frequency and threshold. An intraday frequency reveals more detail about FI interrelationships, compared to a daily frequency. Nevertheless, it is also possible that a daily frequency, in some cases, provides a more-relevant and less-arbitrary metric. A day is a clearly divided unit of time and a basis for the payment cycle.
An arbitrary threshold is an incomplete substitute for measures that take a more complete account of the weights of different links, but it still helps to filter out links that are
unimportant for the network. All bilateral payment values during a sample interval greater than a threshold become links in the network. An intraday frequency also reflects the variability in relationship strength, because, with a higher frequency, links disappear more quickly, unless they recur.
Any thresholds we impose should account for the asymmetry in the amounts that would be significant for different banks. One way to do this uses a participant‘s T2NDC, since this is a rough gauge of the estimation by other banks of the T2NDC recipient‘s credit-bearing capacity. The T2NDC jumps discontinuously, so the threshold uses the mean over the study period. Figure 7 illustrates the distribution of daily non-zero bilateral payment values as a proportion of the smaller T2NDC of the two participants in each bilateral relationship.28 The largest bilateral relationships evidently exist among the big five banks, who are able to use collateral more efficiently, since gross payment values reach a higher multiple of their T2NDCs.29 Links between other banks and the big five are the next most important, followed by the much smaller links among the non-big five banks. This reflects the type of hub-and-spoke network that we typically see in payments systems.
28 We use a 30 per cent system-wide percentage (SWP) throughout the data period, for consistency in the
threshold calculations. BCLs do not change significantly to compensate for the SWP increase from 24 per cent to 30 per cent which occurred on 1 May 2008.
29 It is possible that large-bank thresholds should account for whether they process payments and net out
24
Figure 7: Bilateral Links as a Ratio of the Pair‘s Lower T2NDC
Immediate interpretation of network statistics can be difficult; for example, the
immediacy of network connections alone cannot determine the risk of contagion. Some nodes and links might easily disappear without disruption, especially if the underlying transactions can take place with an alternate counterparty. Other financial relationships, even if less frequent, could involve more risk exposure, and could thus be a source of vulnerability. We could expect, though, that such exposures would more likely exist between FIs that interact regularly, and network distance might affect the speed of contagion; intermediate nodes would first have to be affected before a shock could reach subsequent nodes. A consistently high centrality measure for a certain node might also be an indication of excessive reliance on that node, from a stability or liquidity recycling perspective.
6
Results
Network statistics at the frequency of a day reflect that the LVTS is a small, dense core of the Canadian financial system. Statistics for the financial crisis, from August 2007
onwards, do not vary greatly from statistics for the entire sample period (Figure 9), which points to the stability of the LVTS network and the reliability of the payments system. A 1 per cent threshold for the minimum T2NDC of each pair of banks translates to a daily payment value threshold of close to Can$10 million for an average pair, and
25
but reduces the mean total value of daily payments by only 0.03 per cent. A 40 per cent threshold reduces mean connectivity to 28.9 (± 3.5 standard deviation) per cent, and reduces mean total daily value by 6.9 per cent. Figure 7 shows that such a threshold strongly reduces links among the non-big five banks, but has only a small effect on links among the big five, because there are fewer links between the big five in the left side of the distribution. Table 2 summarizes the main measures for the 1 per cent and 40 per cent thresholds.
Figure 8 shows the effect of a range of thresholds on the means of a few main network measures, which mostly translates to vertical shifts in the lines shown in Figure 9. The threshold of 1 per cent shows the most noticeable incremental change, with connectivity falling by almost 25 per cent and degree centralization jumping by more than threefold. Because the thresholds filter out links mostly among small participants, network
centrality measures reflect a hub-and-spoke characteristic that we might expect in a network with big and small participants.
It would appear that payments in the LVTS tend to be bilaterally coordinated, because reciprocity stays relatively flat across thresholds and averages 89.3 ± 2.5 per cent for the study period (Table 2). Subsequent results use a 1 per cent threshold, unless otherwise noted. The mean clustering coefficient is similarly high, at 84.3 ± 1.5 per cent, compared to other models of PCSSs described in section 3.2. The mean average path length is 1.31 ± 0.03, and the mean average eccentricity is 1.84 ± 0.07. Most of the network also stays connected, even when the threshold increases, as shown by the GSCC. Section 6.1 describes the results for intraday sampling frequencies.
26 Figure 8: Effect of Thresholds on Averages
27
Table 2: Summary of Tranche 2 Network Measures, April 2004–December 2008a
Variable Mean Std. dev. Min Max
(Threshold = 1% of lower T2NDC, no holidays)
Connectivity 0.692 0.032 0.577 0.791 Reciprocity 0.893 0.025 0.803 0.977 AvgPthL 1.308 0.032 1.209 1.423 Diameter 2.005 0.072 2 3 GSCC 13.885 0.319 13 14 Eccentricity 1.841 0.069 1.615 2.143 Clustering 0.843 0.015 0.781 0.883 Betweenness 0.062 0.017 0.023 0.134 Degree 0.358 0.037 0.244 0.481
(Threshold = 40% of lower T2NDC, no holidays)
Connectivity 0.284 0.035 0.038 0.401 Reciprocity 0.793 0.057 0.571 0.967 AvgPthL 1.688 0.066 1.489 2 Diameter 2.718 0.505 2 4 GSCC 12.524 1.033 4 14 Eccentricity 2.184 0.266 1.800 3.385 Clustering 0.573 0.094 0 0.854 Betweenness 0.426 0.093 0.018 0.779 Degree 0.676 0.066 0.135 0.872
(No Threshold, Tranche 2)
Total value 140.99m 23315.42 88600.148 230.49m
Total trans 19650.33 3420.00 13344 38787
Value cent. 0.300 0.017 0.231 0.343
Herfindahl 0.186 0.008 0.163 0.209
a. The Bank of Canada is not included in the network.
With no threshold, visible changes in the statistics include a small downward adjustment of connectivity, and an upward adjustment of the average path length, following the entry of State Street into the LVTS network on 18 October 2004. The effect seems to mostly disappear with a threshold, however, which is surprising in that the new participant does not change the pre-existing pattern. In Figure 9, the shift occurs only with a brief
transition, after which there is a slight trend upwards for connectivity, up until late 2008, probably as financial activity generally increased.
Outliers occur on civic holidays, not shown in Figure 9, where the average path length spikes up to 2.22 and connectivity and reciprocity drop to 16.5 and 33.3 per cent on average, respectively. Diameter for any given day is nearly always two, and the daily GSCC includes all participants, with the exception of civic and U.S. holidays. The spike on 18 February 2008 corresponds to President‘s Day in the United States.
Notwithstanding these outliers, daily network measures test as covariance stationary, although they exhibit significant partial autocorrelations, to varying degrees. In other words, there is a persistence to variations that occur in the network structure in the short term, even while there is never a great departure from the long-run average.
28
Figure 9: Network Statistics (1% Threshold of T2NDC), April 2004–December 2008
Correlations between network statistics tell us how certain network properties relate to one another, possibly leading to ideas about the nature of interbank relationships. In many cases, however, a common construction from the same underlying data could influence a correlation. For example, additional links increase connectivity; if the LVTS is already highly connected, reciprocity would also likely increase, while path lengths would decrease, since additional links add to the density of connections. If the network is sparse, we should expect path lengths to increase and reciprocity to fall as new links connect peripheral nodes that do not yet connect to the GSCC.30 With a daily sampling interval, the former case of high connectivity materializes.
Table 3 summarizes correlations at a daily frequency with a 1 per cent threshold for the 2004–08 sample period.31 Connectivity shows correlations of 0.302 with total value and 0.476 with the total volume of transactions. Value centralization is uncorrelated with network measures at a 1 per cent threshold, including non-weighted centralization measures, but correlates positively with total value and volume. With a 40 per cent threshold, connectivity shows correlations of 0.139 with reciprocity, -0.352 with average
30
These facts remind us of the importance of context in interpreting correlations, which might not be constant with respect to the values of the variables.
31 This table excludes outlying civic holidays and U.S. holidays, which does affect certain correlations, but
29
path length (which still confirms the expected signs of a highly connected network where additional links do not expand the network but, instead, fill in missing links), 0.645 with total value, and 0.584 with total transactions. Even at a 40 per cent threshold,
connectivity negatively correlates with centrality. Reciprocity also correlates positively with total value and volume.
Table 3: Correlations, Excluding Civic and U.S. Holidays, April 2004–December 2008 Conn. Recip. AvgPthL Betw. Valuea Volumea 1% T2NDC Threshold Connectivity 1 Reciprocity 0.37 1 AvgPthL 0.9999 -0.3703 1 Betweenness 0.8052 -0.361 0.8045 1 Total valuea 0.3021 0.2108 -0.3023 0.4381 1 Total volumea 0.4758 0.2631 -0.4758 -0.5253 0.7864 1 Value cent.a 0.0137 -0.0179 0.0142 0.0515 0.2565 0.1737 40% T2NDC Threshold Connectivity 1 Reciprocity 0.1389 1 AvgPthL 0.3523 -0.1715 1 Betweenness -0.184 0.0174 0.3903 1 Total valuea 0.6448 0.2368 -0.2547 0.0689 1 Total volumea 0.5843 0.1512 -0.1928 -0.034 0.7864 1 Value cent.a 0.1299 0.1841 0.0492 0.3193 0.2565 0.1737 a No threshold is imposed.
These correlations confirm our intuition that a higher volume of transactions corresponds to more links in the network, and consequently shorter path lengths. The association of higher total value with more value centralization could relate to higher variability in the activity of the major banks, which account for the majority of value. Higher-value liquidity movement might tend to take place bilaterally, rather than circularly, through a larger number of connected participants; this would involve easier coordination, and is a tentative explanation for the positive correlation of reciprocity and value. The intraday situation provides another case to consider, and reinforces the daily result that indicators of increased payment activity typically increase with one another, when compensating for the time-of-day effect.
Fedwire connectivity is strongly positively correlated with value and volume, while reciprocity is not significantly correlated with value, volume, or connectivity (Soramäki et al. 2006), which points to a case where the bilateral coordination of payments does not vary with these other characteristics, in contrast to the LVTS. In Fedwire, reciprocity also correlates negatively with network size, possibly because non-reciprocal links might often expand the network.
